A brief introduction to quasiparticles in frustrated magnets - PowerPoint PPT Presentation

A brief introduction to quasiparticles in frustrated magnets Claudio Castelnovo TCM group Cavendish Laboratory University of Cambridge 17-06-2020 CMP in All the Cities

Outline ◮ magnetism, frustration and spin liquid behaviour ◮ modelling spin liquids: general overview ◮ quasiparticle excitations: 6-vertex and 8-vertex model • classical behaviour: deconfinement, fractionalisation, dynamical constraints and entropic interactions • quantum behaviour: fractional statistics and dual quasiparticles → toric code and quantum spin ice ◮ quantum spin liquids at finite temperature (a prelude to the second talk) ◮ conclusions

Conventional Magnetism vs Spin Liquids “trivial” disorder order T / J ~ 1 “trivial”: T � J ⇒ high- T expansion holds ( � S i S j � ∼ − H ij / T ) frustration: inability to minimise locally all energy terms ⇒ T c ≪ J H = J � ij S i S j (triang. Ising AFM)

Conventional Magnetism vs Spin Liquids “trivial” disorder order T / J ~ 1 “trivial”: T � J ⇒ high- T expansion holds ( � S i S j � ∼ − H ij / T ) frustration: inability to minimise locally all energy terms ⇒ T c ≪ J ? order disorder T / J ~ 1 T /J << 1 c

Conventional Magnetism vs Spin Liquids ? order disorder T / J ~ 1 T /J << 1 c ◮ ∼ 1 is typically a crossover (Schottky anomaly) ◮ no long range order ◮ non-trivial spin correlations ( T < J ⇒ � S i S j � ∼ − H ij / T ) − → spin liquid

Modelling (classical) spin liquids example: nn Ising AFM on triangular lattice 2:1 triangles vs 3:0 triangles energy difference: ∆ ∼ J ⇒ projects onto mostly 2:1 configurations for T � ∆

Modelling (classical) spin liquids generally: H ∼ H ∆ + H δ ◮ leading contribution ( H ∆ ) projects onto subset of configuration space (no spontaneous symmetry breaking) for T � ∆ ◮ possible subleading contributions ( H δ ) cause ordering for T � δ ≪ ∆ (triang. nn Ising AFM: H δ = 0) SL order disorder d D T / / << 1 ~ 1 D

Effective dimer description � for T � ∆, mostly 2:1 triangles � ferro bonds equivalent to dimers on dual honeycomb lattice leading to: ◮ extensive degeneracy ◮ non-trivial correlations

Emergent gauge symmetry and dipolar correlations � dimer = flux 2 from A to B B � no-dimer = flux 1 from B to A A � dimer constraint = divergenceless condition − → emergent gauge field Henley AR 2010 ⇒ 2D dipolar correlations: � flux flux � ∼ � dimer dimer � ∼ � S S �

Elementary excitations of H ∆ - + for convenience: + + + + B A Ising model on + + bonds of square - + - A B lattice + - consider Hamiltonians that result in leading projection term H ∆ favouring: i - + + s + + - ◮ � i ∈ s σ i = 0 - - (6 vertex model) i - + + ◮ � i ∈ s σ i = 1 (8 vertex model) + s + + - + + - - +

Six-vertex vs eight-vertex model ◮ extensively degenerate Pauling’s entropy estimate: 2 N spins, N sites n out of 16 ( n = 6 , 8) minimal energy configurations per site 2 2 N � n � � N � S ∼ ln ∼ s n N 16 ◮ unusual correlations 6-vertex model: σ i = ± 1 ⇔ flux from A to B (B to A) ⇒ divergenceless condition and dipolar correlations [Isakov PRL 2004] 8-vertex model: plaquette flips preserve minimal energy ⇒ zero-range corr. � σ i σ j � = 0, ∀ i � = j but topological properties

Excitations in the 8-vertex model - + + + - - - + - spin flip on ground state + - - - + + + - + ⇓ - two defects: � i ∈ s σ i = − 1 + + + - - + + - + + + - ◮ spins next to defect flip at no energy cost (hop or annihilate) ◮ trivially deconfined ◮ elementary excitations are single defective sites ◮ lattice gas of (RW) particles that pair create or annihilate

Excitations in the 8-vertex model - + + + - - - + - spin flip on ground state + - - - + - + - + ⇓ - two defects: � i ∈ s σ i = − 1 + + + - - + + - + + + - ◮ spins next to defect flip at no energy cost (hop or annihilate) ◮ trivially deconfined ◮ elementary excitations are single defective sites ◮ lattice gas of (RW) particles that pair create or annihilate

Excitations in the 8-vertex model - + + + - - - + - spin flip on ground state + - - - + - + - + ⇓ + two defects: � i ∈ s σ i = − 1 + + + - - + + - + + + - ◮ spins next to defect flip at no energy cost (hop or annihilate) ◮ trivially deconfined ◮ elementary excitations are single defective sites ◮ lattice gas of (RW) particles that pair create or annihilate

Excitations in the 8-vertex model - + + + - - - + - spin flip on ground state + - - + + - + - + ⇓ - + two defects: � i ∈ s σ i = − 1 + + - - - + - + + + - ◮ spins next to defect flip at no energy cost (hop or annihilate) ◮ trivially deconfined ◮ elementary excitations are single defective sites ◮ lattice gas of (RW) particles that pair create or annihilate

Excitations in the 6-vertex model - + + + - - + + - spin flip on ground state + - - - - - + - + ⇓ two defects: � i ∈ s σ i = ± 1 + + + + - - - + - + + - - ◮ spins next to defect flip at no energy cost only along alternating sign paths (+ − + − + − . . . ) ◮ elementary excitations are single defective sites (deconfined) ◮ constrained gas of particles that pair create or annihilate

Excitations in the 6-vertex model - + + + - - + + - spin flip on ground state + - - - - + + - + ⇓ two defects: � i ∈ s σ i = ± 1 + + + + - - - + - + + - - ◮ spins next to defect flip at no energy cost only along alternating sign paths (+ − + − + − . . . ) ◮ elementary excitations are single defective sites (deconfined) ◮ constrained gas of particles that pair create or annihilate

Excitations in the 6-vertex model - + + + - - + + - spin flip on ground state + - - - + - + - + ⇓ - + + + two defects: � i ∈ s σ i = ± 1 - - - + - + + - - ◮ spins next to defect flip at no energy cost only along alternating sign paths (+ − + − + − . . . ) ◮ elementary excitations are single defective sites (deconfined) ◮ constrained gas of particles that pair create or annihilate

Excitations in the 6-vertex model - + + + - - + + - spin flip on ground state + - - + - + + - + ⇓ - - + + two defects: � i ∈ s σ i = ± 1 - - + + - + + - - ◮ spins next to defect flip at no energy cost only along alternating sign paths (+ − + − + − . . . ) ◮ elementary excitations are single defective sites (deconfined) ◮ constrained gas of particles that pair create or annihilate

Excitations in the 6-vertex model (gauge flux rep.) spin flip on ground state ⇓ pair of oppositely charged defects: sinks (3i1o) and sources (3o1i) of gauge flux ◮ defects move freely along oriented arrow paths ◮ close interplay: spins determine how defects move; defect motion rearranges spins ◮ lattice gas of gauge charges → spins mediate Coulomb int.

Excitations in the 6-vertex model (gauge flux rep.) spin flip on ground state ⇓ pair of oppositely charged defects: sinks (3i1o) and sources (3o1i) of gauge flux ◮ defects move freely along oriented arrow paths ◮ close interplay: spins determine how defects move; defect motion rearranges spins ◮ lattice gas of gauge charges → spins mediate Coulomb int.

Excitations in the 6-vertex model (gauge flux rep.) spin flip on ground state ⇓ pair of oppositely charged defects: sinks (3i1o) and sources (3o1i) of gauge flux ◮ defects move freely along oriented arrow paths ◮ close interplay: spins determine how defects move; defect motion rearranges spins ◮ lattice gas of gauge charges → spins mediate Coulomb int.

Excitations in the 6-vertex model (gauge flux rep.) spin flip on ground state ⇓ pair of oppositely charged defects: sinks (3i1o) and sources (3o1i) of gauge flux ◮ defects move freely along oriented arrow paths ◮ close interplay: spins determine how defects move; defect motion rearranges spins ◮ lattice gas of gauge charges → spins mediate Coulomb int.

Parenthesis: entropic Coulomb interaction ◮ no energetic interactions between defects ◮ yet probability P ( R ) of two oppositely charged defects R apart ∼ exp[ C d ( R )] with Coulomb potential C d ( R ) in d dim. ◮ ⇒ entropic Coulomb interaction − T C d ( R ) − 2.45 10 distribution function − 2.48 10 − 2.51 10 − 2.54 10 0 0.1 0.2 0.3 0.4 inverse monopole separation (units of pyrochlore a) d = 3, C d ( R ) ∼ 1 / R CC et al. PRB 2011

Low temperature (classical) dynamics behaviour controlled by sparse defect motion: ◮ 8-vertex: 2D random walk + pair creation/annihilation events (aka reaction-diffusion process) ◮ 6-vertex: constrained lattice gas motion + entropic Coulomb interactions Toussaint et al. J. Chem. Phys. 1983 Ginzburg et al. PRE 1997 Ryzhkin et al. JETP 2005, EPL 2013 CC et al. PRL 2010, PRB 2019

Quantum spin liquids H = H ∆ + H δ where H δ ∼ H defect int. + H defect hopping ◮ neglect H defect int. for simplicity ◮ hopping t � ∆ → defect dynamics (first order) + ‘ground state’ dynamics (perturbatively: ∆ ∼ t ( t / ∆) n ) D ring exchange